球面码

不同维度球面上的点的优美排列

斯隆

电子邮件地址:NJasLoa@ gmail .com主页:尼尔斯兰

基于与R. H. Hardin、W. D. Smith等人的合作

 一个假定最优球码的库,连同其他的排列可能不是最优的,但由于某些原因特别有趣。

 关键词points on a sphere, spherical codes, spherical designs, spherical coverings, 4-dimensional arrangements, 5-dimensional arrangements, 6-dimensional arrangements, 7-dimensional arrangements, 8-dimensional arrangements, 9-dimensional arrangements, 10-dimensional arrangements, 11-dimensional arrangements, 12-dimensional arrangements, 13-dimensional arrangements, 14-dimensional arrangements, 15-dimensional arrangements, 16-dimensional arrangements, 17-dimensional arrangements, 18-dimensional arrangements, 19-dimensional arrangements, 20-dimensional arrangements, 21-dimensional arrangements, 22-dimensional arrangements, 23-dimensional arrangements, 24-dimensional arrangements, higher-dimensional arrangements, Archimedian solids, Barnes-Wall lattice, Coxeter-Todd lattice, cube, cuboctahedron, dodecahedron, E_8 lattice, Gosset lattice, hypercube, hypersphere, icosahedron, Leech lattice, Nordstrom-Robinson code, octahedron, generalized octahedron, orthoplex, Platonic solids, rhombic dodecahedron, simplex, 600-cell, sNub立方体、改进的Snbb立方体、四面体、截截立方体、截头十二面体、截头二十面体、截短八面体、截短四面体、24单元、扭立方体

 问题将n个点放置在球面上的D维上,以使它们之间的最小距离(或等效最小角度)最大化。

This file has two parts

  1. 第1部分在3, 4和5维中假定最佳填料的表,n=4,…,130点。
  2. 第2部分其他良好(虽然并不总是最佳)的填料在尺寸3到24之间的填料,具有不同数量的点。

评论


第1部分,尺寸3到5,最多130点

结果总结:

球面上的点的Packings

没有其他来源被列出的安排是版权R. H. Hardin,N.J.A.斯隆和W.D史密斯,1994年至1996年。

dim npts min separation (degrees)

  3    4  109.4712206
  3    5   90.0000000
  3    6   90.0000000
  3    7   77.8695421
  3    8   74.8584922
  3    9   70.5287794
  3   10   66.1468220
  3   11   63.4349488
  3   12   63.4349488
  3   13   57.1367031
  3   14   55.6705700
  3   15   53.6578501
  3   16   52.2443957
  3   17   51.0903285
  3   18   49.5566548
  3   19   47.6919141
  3   20   47.4310362
  3   21   45.6132231	ref D.A. Kottwitz (Acta Cryst., A47, 158-165(1991)) thanks to Jim Buddenhagen (jbuddenh(AT)earthlink.net)
  3   22   44.7401612
  3   23   43.7099642
  3   24   43.6907671
  3   25   41.6344612
  3   26   41.0376616
  3   27   40.6776007	ref D.A. Kottwitz (Acta Cryst., A47, 158-165(1991)) thanks to Jim Buddenhagen (jbuddenh(AT)earthlink.net)
  3   28   39.3551436
  3   29   38.7136512
  3   30   38.5971159
  3   31   37.7098291
  3   32   37.4752140
  3   33   36.2545530	ref D.A. Kottwitz (Acta Cryst., A47, 158-165(1991)) thanks to Jim Buddenhagen (jbuddenh(AT)earthlink.net)
  3   34   35.8077844
  3   35   35.3198076
  3   36   35.1897322
  3   37   34.4224080
  3   38   34.2506607
  3   39   33.4890466
  3   40   33.1583563
  3   41   32.7290944
  3   42   32.5063863
  3   43   32.0906244
  3   44   31.9834230
  3   45   31.3230814
  3   46   30.9591635
  3   47   30.7818159
  3   48   30.7627855
  3   49   29.9235851
  3   50   29.7529564
  3   51   29.3684069
  3   52   29.1947579
  3   53   28.8138972
  3   54   28.7169205	ref D.A. Kottwitz (Acta Cryst., A47, 158-165(1991)) thanks to Jim Buddenhagen (jbuddenh(AT)earthlink.net)
  3   55   28.2627914
  3   56   28.1480466
  3   57   27.8266759
  3   58   27.5564159
  3   59   27.3949757
  3   60   27.1928300
  3   61   26.8732779
  3   62   26.6839970
  3   63   26.4869225
  3   64   26.2350433
  3   65   26.0698299
  3   66   25.9474437	ref D.A. Kottwitz (Acta Cryst., A47, 158-165(1991)) thanks to Jim Buddenhagen (jbuddenh(AT)earthlink.net)
  3   67   25.6839813
  3   68   25.4638245
  3   69   25.3336364	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
  3   70   25.1709200
  3   71   24.9879381
  3   72   24.9264861
  3   73   24.5537792	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
  3   74   24.4209398	D.A. Kottwitz, private communication
  3   75   24.3017225
  3   76   24.1281944
  3   77   24.0012837
  3   78   23.9310254
  3   79   23.6239917
  3   80   23.5530672	Jim Buddenhagen (jbuddenh(AT)earthlink.net) private communication
  3   81   23.3476377	D.A. Kottwitz, private communication
  3   82   23.1946074
  3   83   23.0829976
  3   84   23.0517306
  3   85   22.7791621
  3   86   22.6743694	ref D.A. Kottwitz (Acta Cryst., A47, 158-165(1991)) thanks to Jim Buddenhagen (jbuddenh(AT)earthlink.net)
  3   87   22.5466574
  3   88   22.4678810
  3   89   22.3166023
  3   90   22.1540232
  3   91   22.0517963
  3   92   22.0275815
  3   93   21.8103801
  3   94   21.7237135	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
  3   95   21.5945501
  3   96   21.5206099
  3   97   21.4006197
  3   98   21.3710607
  3   99   21.1359674	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
  3  100   21.0312020
  3  101   20.9286834
  3  102   20.8556887	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
  3  103   20.7382700	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
  3  104   20.6566210
  3  105   20.5388524
  3  106   20.4394089
  3  107   20.3612035	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
  3  108   20.3044447	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
  3  109   20.1493196	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
  3  110   20.1113276	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
  3  111   19.9824769	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
  3  112   19.8913044	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
  3  113   19.8056013
  3  114   19.7450093
  3  115   19.6239931	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
  3  116   19.5497969	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
  3  117   19.4612911
  3  118   19.3893497	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
  3  119   19.3257514	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
  3  120   19.3240201
  3  121   19.1357298	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
  3  122   19.0700369	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
  3  123   19.0063891	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
  3  124   18.9539116	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
  3  125   18.8448151	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
  3  126   18.7815856
  3  127   18.6900568	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
  3  128   18.6349726
  3  129   18.5634726	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
  3  130   18.5103522
  4    5  104.4775121
  4    6   90.0000000
  4    7   90.0000000
  4    8   90.0000000
  4    9   80.6761150
  4   10   80.4059318
  4   11   76.6790498
  4   12   75.5224878
  4   13   72.1036706
  4   14   71.3661859
  4   15   69.4519870
  4   16   67.1930021
  4   17   65.6531563
  4   18   64.9872827
  4   19   64.2618766
  4   20   64.2618766
  4   21   61.8760316
  4   22   60.1398863
  4   23   60.0000000
  4   24   60.0000000
  4   25   57.4988826
  4   26   57.2625923
  4   27   56.1106132
  4   28   55.4351398
  4   29   55.0299245
  4   30   54.2511897
  4   31   53.7886221
  4   32   53.4360130
  4   33   53.0759811
  4   34   52.4011334
  4   35   51.5724903
  4   36   51.0745199
  4   37   50.9226685
  4   38   50.1535207
  4   39   49.6895967
  4   40   49.4208547
  4   41   49.0594727
  4   42   48.8204491
  4   43   48.4963928
  4   44   48.0251062
  4   45   47.3961868
  4   46   46.9743371
  4   47   46.7328543
  4   48   46.3832527
  4   49   46.0990591
  4   50   46.0742073
  4   51   45.4625355
  4   52   45.2812081
  4   53   44.9542243
  4   54   44.7883187
  4   55   44.3025299
  4   56   44.0114102
  4   57   43.7495816
  4   58   43.5341667
  4   59   43.3907742
  4   60   43.2167495
  4   61   42.9862214
  4   62   42.7870922
  4   63   42.3770430
  4   64   42.3062196
  4   65   42.0566173
  4   66   41.7550028
  4   67   41.4987735
  4   68   41.3146370
  4   69   41.1813742
  4   70   41.0851193
  4   71   40.8538202
  4   72   40.7098566
  4   73   40.6070686
  4   74   40.5754396
  4   75   40.4865078
  4   76   40.4418535
  4   77   40.0749825
  4   78   39.7568300
  4   79   39.3704881
  4   80   39.1321954
  4   81   38.9404678
  4   82   38.7616584
  4   83   38.6411550
  4   84   38.5216197
  4   85   38.3971687
  4   86   38.2834796
  4   87   38.2316167
  4   88   38.1934424
  4   89   37.9525738
  4   90   37.7779155
  4   91   37.6840939
  4   92   37.6073513
  4   93   37.5288195
  4   94   37.4811292
  4   95   37.1570600
  4   96   37.0622214
  4   97   36.9191853
  4   98   36.7966891
  4   99   36.6983263
  4  100   36.5636318
  4  101   36.4956325
  4  102   36.4186254
  4  103   36.3807276
  4  104   36.2615258
  4  105   36.1859494
  4  106   36.1333759
  4  107   36.0959131
  4  108   36.0509369
  4  109   36.0309659
  4  110   36.0076295
  4  111   36.0013484
  4  112   36.0000058
  4  113   36.0000000
  4  114   36.0000000
  4  115   36.0000000
  4  116   36.0000000
  4  117   36.0000000
  4  118   36.0000000
  4  119   36.0000000
  4  120   36.0000000
  4  121   34.4465422
  4  122   34.2716955
  4  123   34.1743519
  4  124   33.9853008
  4  125   33.8729785
  4  126   33.7679953
  4  127   33.6685430
  4  128   33.5726406
  4  129   33.5033828
  4  130   33.4075174
  4  600   19.8433177
  5    6  101.5369590
  5    7   90.0000000
  5    8   90.0000000
  5    9   90.0000000
  5   10   90.0000000
  5   11   82.3654810
  5   12   81.1451619
  5   13   79.2070847
  5   14   78.4630410
  5   15   78.4630410
  5   16   78.4630410
  5   17   74.3074162
  5   18   74.0080831
  5   19   73.0329080
  5   20   72.5792898
  5   21   71.6441994
  5   22   69.2068599
  5   23   68.2984004
  5   24   68.0230856
  5   25   67.6897899
  5   26   67.0305252
  5   27   66.3175208
  5   28   65.9100901
  5   29   65.7301670
  5   30   65.6046967
  5   31   64.2839257
  5   32   63.7779969
  5   33   62.6125998
  5   34   61.9343521
  5   35   61.4165697
  5   36   61.0530896
  5   37   60.4074688
  5   38   60.2482021
  5   39   60.0017727
  5   40   60.0000000
  5   41   59.0029024
  5   42   58.7856545
  5   43   58.3513041
  5   44   58.1661569
  5   45   57.6514221
  5   46   57.4326036
  5   47   57.1800994
  5   48   56.9676480
  5   49   56.8441582
  5   50   56.8441582
  5   51   56.3071628
  5   52   55.9343732
  5   53   55.4667168
  5   54   55.1747097
  5   55   54.9279561
  5   56   54.7453222
  5   57   54.7356103
  5   58   54.4462372
  5   59   54.0539784
  5   60   53.6051457
  5   61   53.3979587
  5   62   53.2010567
  5   63   52.8751126
  5   64   52.7108774
  5   65   52.5214037
  5   66   52.4752830
  5   67   52.2729583
  5   68   52.0286818
  5   69   51.9176291
  5   70   51.8973507
  5   71   51.8317602
  5   72   51.8317602
  5   73   51.8272924
  5   74   51.8272924
  5   75   51.3428135
  5   76   50.8459281
  5   77   50.4692075
  5   78   50.1905273
  5   79   49.9276863
  5   80   49.6870598
  5   81   49.5416542
  5   82   49.3488655
  5   83   49.0496962
  5   84   48.8819546
  5   85   48.7570741
  5   86   48.6290153
  5   87   48.5000839
  5   88   48.3577476
  5   89   48.2141763
  5   90   48.1896851
  5   α,α,ε,α,β,α,β,α,β,α,β,α,β,α,β,β,α,β,β,α,β,β,α,β,β,α,β,β,β,β,α,β,β,α,β,β,α,β,α,β,α,β,β,α,β,β,α,β,β,α,β,β,α,β,β,α,β,β,α,β,α,β,β,α,β,α,β,α,β,β,α,β,α,β,α,β,β,α,β,α,β,α,β,β,α,β,α,β,α,β,α,β,α,β,α,β,α,β,α,β,α,β,α,β,α,β,α,β,α,β,α,β,α,β,α,β,α,β,α,β,α,β,α,β,α,β,α,β,α,β,α,β,α,β,α,β,α,β,α,β,α,β,α,β,α,β,α,β,α,β,α,β,α,β,α,β,α,β,α,β,α,β,α,β,α,β,α,β,α,β,α,β,α,β,α,β,α,β,α,β,α,β,α,β,α,β,α,β,α,β,α,β,α,β,α,β,α,β,α,β,α,β,α,β,α,β,α,β,α,β,α,β,α,β,α,β,α,β,α,β,α,β,α,β,β91、47.9008875、5、92、47.7716828、5、93、47.6364239、5、94、47.5270717、47.5270717、5、47.5270717、α、α、β、ε、β、ε之间的关系;


第2部分,球面上点的其它优美排列

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